initial

other/elliptic_curve.py

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+#!/usr/bin/env python3
+try:
+    from PIL import Image, ImageDraw
+    IMG_OUT = True
+except ModuleNotFoundError:
+    IMG_OUT = False
+
+
+def power(x, y, p):
+    res = 1
+    x %= p
+    while (y > 0):
+        if (y & 1):
+            res = (res * x) % p
+        y = y >> 1
+        x = (x * x) % p
+    return res
+
+
+# Assumption: p is of the form 3*i + 4 where i >= 1
+def sqrtFast(n, p):
+    if (p % 4 != 3):
+        raise ValueError('Invalid Input')
+    # Try "+(n ^ ((p + 1)/4))"
+    n = n % p
+    x = power(n, (p + 1) // 4, p)
+    if ((x * x) % p == n):
+        return x
+    # Try "-(n ^ ((p + 1)/4))"
+    x = p - x
+    if ((x * x) % p == n):
+        return x
+    return None
+
+
+def sqrtNormal(n, p):
+    n %= p
+    for x in range(2, p):
+        if ((x * x) % p == n):
+            return x
+    return None
+
+
+def elliptic_curve(a, b, r):
+    print(f'generate curve: a={a}, b={b}, r={r}')
+    if IMG_OUT:
+        image1 = Image.new('RGB', (r, r))
+        draw1 = ImageDraw.Draw(image1)
+        draw1.rectangle((0, 0, r, r), fill='white')
+        image2 = Image.new('RGB', (r, r))
+        draw2 = ImageDraw.Draw(image2)
+        draw2.rectangle((0, 0, r, r), fill='white')
+
+    sqrtFn = sqrtNormal if (r % 4 != 3) else sqrtFast
+    txt = ''
+    for x in range(r):
+        y2 = (x ** 3 + a * x + b) % r
+        u2 = sqrtFn(y2, r) if y2 > 0 else 0
+        if u2 is not None:
+            z1 = r - 1 - u2
+            z2 = r - 1 - (-u2 % r)
+            print(x, y2, -y2 % r)
+            txt += f'{x} {y2} {-y2 % r}\n'
+            if IMG_OUT:
+                draw1.rectangle((x, z1, x, z1), fill='black')
+                draw1.rectangle((x, z2, x, z2), fill='black')
+                draw2.rectangle((x - 2, z1 - 2, x + 2, z1 + 2), fill='black')
+                draw2.rectangle((x - 2, z2 - 2, x + 2, z2 + 2), fill='black')
+
+    with open(f'ec-a{a}-b{b}-r{r}.txt', 'w') as f:
+        f.write(txt)
+    if IMG_OUT:
+        print('writing image output')
+        image1.save(f'ec-a{a}-b{b}-r{r}-pp.png', 'PNG')
+        image2.save(f'ec-a{a}-b{b}-r{r}-lg.png', 'PNG')
+    print()
+
+
+elliptic_curve(a=149, b=263, r=3299)